To ensure that the map $X \to \mathbb{P}^1$ provided is indeed a Belyi map, we first verify by taking (exact) preimages that the ramification above $\{0,1,\infty\} \subseteq \mathbb{P}^1$ is correct (in particular, it has the correct degree). We then also verify that $X$ has the correct genus; it follows by the Riemann-Hurwitz theorem that there can be no other ramification, so the map is indeed a Belyi map.
The complex accuracy set in the numerical triangle group method is undoubtedly enough then to automatically ensure that the given map induces the described permutation triple up to simultaneous conjugation, namely via its action by monodromy on the preimages. This is very unlikely to be incorrect.
Once the Belyi map is verified and associated with the correct permutation triple for all Belyi maps of a given degree, base fields and Galois orbits can be read off directly. Other invariants of the map or its orbits can then be obtained from the corresponding monodromy triple by means of a short calculation.